Exact Algorithms With Worst-case Guarantee For Scheduling: From Theory to Practice. (Algorithmes Exacts Avec Garantie au Pire Cas Pour l'Ordonnancement: de la Théorie à la Pratique)

This thesis summarizes the author’s PhD research works on the design of exact algorithms that provide a worst-case (time or space) guarantee for NP-hard scheduling problems. Both theoretical and practical aspects are considered with three main results reported. The first one is about a Dynamic Programming algorithm which solves the F3||Cmax problem in O∗(3n) time and space. The algorithm is easily generalized to other flowshop problems including F2||f and F3||f , and single machine scheduling problems like 1|ri|f , with f ∈ {fmax, fi}. The second contribution is about a search tree method called Branch & Merge which solves the 1||Ti problem with the time complexity converging to O∗(2n) and in polynomial space. The work is based on the observation that many identical subproblems appear during the solution of the input problem. An operation called merge is then derived, which merges all identical nodes to one whenever possible and hence yields a better complexity. Our third contributionaims to improve the practical efficiency of exact search tree algorithms solving scheduling problems. First we realized that a better way to implement the idea of Branch & Merge is to use a technique called Memorization. By the finding of a new algorithmic paradox and the implementation of a memory cleaning strategy, the method succeeded to solve instances with 300 more jobs with respect to the state-of-the-art algorithm for the 1||Ti problem. Then the treatment is extended to another three problems 1|ri| ∑ Ci, 1|d̃i| ∑ wiCi and F2|| ∑ Ci previously addressed by T’kindt et al. (2004). The results of the four problems all together show the power of the Memorization paradigm when applied on sequencing problems. We name it Branch & Memorize to promote a systematic consideration of Memorization as an essential building block in branching algorithms like Branch & Bound. The method can surely also be used to solve other problems, which are not necessarily scheduling problems.